= Background gauge-field renormalization identity
{title2=$Z_gZ_A^{1/2}=1$}
For a background <gauge field> with $A_0=Z_A^{1/2}A$ and $g_0=\mu^{\epsilon/2}Z_gg$, background gauge covariance preserves the combination $gA$ in the <gauge covariant derivative>. Its <Ward identity> therefore gives the displayed relation. This refers to background-field <renormalization>, not an unrestricted identity between the coupling and the ordinary quantum-gluon wave-function factor. Gauge-invariant one-loop matter determinants can be renormalized with this identity to extract their coupling running from vacuum polarization.
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